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Solve -3x² + 8x - 4 = 0

The solutions of -3x² + 8x - 4 = 0 are x = 2/3 or x = 2.

x² +x += 0
−3x2+8x−4=0⇒x=23, 2-3x^2 + 8x - 4 = 0 \Rightarrow x = \frac{2}{3},\ 2
Discriminant D = 16 · x ≈ 0.666667, 2

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

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Graph of y = -3x² + 8x - 4-11234-30-25-20-15-10-55x = 0.66666667x = 2vertexy = -3x² + 8x - 4
Graph of y = -3x² + 8x - 4. The curve crosses the x-axis at x = 0.66666667 and x = 2; the vertex is at (1.3333333, 1.3333333).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -3, b = 8, c = -4.

    −3x2+8x−4=0-3x^2 + 8x - 4 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=82−4⋅(−3)⋅(−4)=64−48=16D = 8^2 - 4\cdot(-3)\cdot(-4) = 64 - 48 = 16
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    −(3x−2)(x−2)=0-\left(3x - 2\right)\left(x - 2\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    3x−2=0⇒x=23x−2=0⇒x=23x - 2 = 0 \Rightarrow x = \frac{2}{3}\qquad x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−8±16−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{16}}{-6}
  7. 7

    Simplify to get the solutions:

    x=23orx=2x = \frac{2}{3}\quad\text{or}\quad x = 2
  8. 8

    Check by putting x = 2/3 back in: a·x² + b·x + c = -3·(4/9) + 8·(2/3) + (-4) = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 8x - 4 = 0\ \Rightarrow\ x = \frac{2}{3},\ 2 \]
\[ -3x^2 + 8x - 4 = 0 \]
\[ D = 8^2 - 4\cdot(-3)\cdot(-4) = 64 - 48 = 16 \]
\[ -\left(3x - 2\right)\left(x - 2\right) = 0 \]
\[ 3x - 2 = 0 \Rightarrow x = \frac{2}{3}\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{16}}{-6} \]
\[ x = \frac{2}{3}\quad\text{or}\quad x = 2 \]

The parabola y = -3x² + 8x - 4

xy
-1.6667-25.667267
-0.6667-10.667067
0.3333-1.6668667
1.33331.3333333
2.3333-1.6664667
3.3333-10.666267
4.3333-25.666067

Questions

What are the solutions of -3x² + 8x - 4 = 0?

The solutions of -3x² + 8x - 4 = 0 are x = 2/3 or x = 2.

Does -3x² + 8x - 4 = 0 factor?

Yes: -1(3x-2)(x-2) = 0.

What is the vertex of y = -3x² + 8x - 4?

(4/3, 4/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 8/3, product = c/a = 4/3.

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